Detecting periodicity with horizontal visibility graphs

The horizontal visibility algorithm was recently introduced as a mapping between time series and networks. The challenge lies in characterizing the structure of time series (and the processes that generated those series) using the powerful tools of graph theory. Recent works have shown that the visibility graphs inherit several degrees of correlations from their associated series, and therefore such graph theoretical characterization is in principle possible. However, both the mathematical grounding of this promising theory and its applications are in its infancy. Following this line, here we address the question of detecting hidden periodicity in series polluted with a certain amount of noise. We first put forward some generic properties of horizontal visibility graphs which allow us to define a (graph theoretical) noise reduction filter. Accordingly, we evaluate its performance for the task of calculating the period of noisy periodic signals, and compare our results with standard time domain (autocorrelation) methods. Finally, potentials, limitations and applications are discussed.

Modeling the evolution of item rating networks using time-domain preferential attachment

The understanding of the structure and dynamics of the intricate network of connections among people that consumes products through Internet appears as an extremely useful asset in order to study emergent properties related to social behavior. This knowledge could be useful, for example, to improve the performance of personal recommendation algorithms. In this contribution, we analyzed five-year records of movie-rating transactions provided by Netflix, a movie rental platform where users rate movies from an online catalog. This dataset can be studied as a bipartite user-item network whose structure evolves in time. Even though several topological properties from subsets of this bipartite network have been reported with a model that combines random and preferential attachment mechanisms [Beguerisse Díaz et al., 2010], there are still many aspects worth to be explored, as they are connected to relevant phenomena underlying the evolution of the network. In this work, we test the hypothesis that bursty human behavior is essential in order to describe how a bipartite user-item network evolves in time. To that end, we propose a novel model that combines, for user nodes, a network growth prescription based on a preferential attachment mechanism acting not only in the topological domain (i.e. based on node degrees) but also in time domain. In the case of items, the model mixes degree preferential attachment and random selection. With these ingredients, the model is not only able to reproduce the asymptotic degree distribution, but also shows an excellent agreement with the Netflix data in several time-dependent topological properties.

Nonlocal analysis of modular roles

We introduce a new methodology to characterize the role that a given node plays inside the community structure of a complex network. Our method relies on the ability of the links to reduce the number of steps between two nodes in the network, which is measured by the number of shortest paths crossing each link, and its impact on the node proximity. In this way, we use node closeness to quantify the importance of a node inside its community. At the same time, we define a participation coefficient that depends on the shortest paths contained in the links that connect two communities. The combination of both parameters allows to identify the role played by the nodes in the network, following the same guidelines introduced by Guimerà et al. [Guimerà & Amaral, 2005] but, in this case, considering global information about the network. Finally, we give some examples of the hub characterization in real networks and compare our results with the parameters most used in the literature.

Digital chaotic output from an optically processing element

Digital chaotic behavior in an optically processing element is reported. It is obtained as the result of processing two fixed trains of bits. The process is performed with an optically programmable logic gate, previously reported as a possible main block for optical computing. Outputs for some specific conditions of the circuit are given. Digital chaos is obtained using a feedback configuration. Period doublings in a Feigenbaum‐like scenario are obtained. A new method to characterize this type of digital chaos is reported.

Damping models in the truncated derivative nonlinear Schrödinger equation

Four-dimensional flow in the phase space of three amplitudes of circularly polarized Alfven waves and one relative phase, resulting from a resonant three-wave truncation of the derivative nonlinear Schrödinger equation, has been analyzed; wave 1 is linearly unstable with growth rate , and waves 2 and 3 are stable with damping 2 and 3, respectively. The dependence of gross dynamical features on the damping model as characterized by the relation between damping and wave-vector ratios, 2 /3, k2 /k3, and the polarization of the waves, is discussed; two damping models, Landau k and resistive k2, are studied in depth. Very complex dynamics, such as multiple blue sky catastrophes and chaotic attractors arising from Feigenbaum sequences, and explosive bifurcations involving Intermittency-I chaos, are shown to be associated with the existence and loss of stability of certain fixed point P of the flow. Independently of the damping model, P may only exist as against flow contraction just requiring.In the case of right-hand RH polarization, point P may exist for all models other than Landau damping; for the resistive model, P may exist for RH polarization only if 2+3/2.

Fully 3-wave model to study the hard transition to chaotic dynamics in alfven wave-fronts

The derivative nonlinear Schrödinger (DNLS) equation, describing propagation of circularly polarized Alfven waves of finite amplitude in a cold plasma, is truncated to explore the coherent, weakly nonlinear coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. No matter how small the growth rate of the unstable wave, the four-dimensional flow for the three wave amplitudes and a relative phase, with both resistive damping and linear Landau damping, exhibits chaotic relaxation oscillations that are absent for zero growth-rate. This hard transition in phase-space behavior occurs for left-hand (LH) polarized waves, paralleling the known fact that only LH time-harmonic solutions of the DNLS equation are modulationally unstable. The parameter domain developing chaos is much broader than the corresponding domain in a reduced 3-wave model that assumes equal dampings of the daughter waves

Decomposition driven interface evolution for layers of binary mixtures: I. Model dirivation and base states

Algebraic topology (homology) is used to analyze the state of spiral defect chaos in both laboratory experiments and numerical simulations of Rayleigh-Bénard convection. The analysis reveals topological asymmetries that arise when non-Boussinesq effects are present. The asymmetries are found in different flow fields in the simulations and are robust to substantial alterations to flow visualization conditions in the experiment. However, the asymmetries are not observable using conventional statistical measures. These results suggest homology may provide a new and general approach for connecting spatiotemporal observations of chaotic or turbulent patterns to theoretical models.

Multiple attractors, long chaotic transients, and failure in small-world networks of excitable neurons.

We study the dynamical states of a small-world network of recurrently coupled excitable neurons, through both numerical and analytical methods. The dynamics of this system depend mostly on both the number of long-range connections or ?shortcuts?, and the delay associated with neuronal interactions. We find that persistent activity emerges at low density of shortcuts, and that the system undergoes a transition to failure as their density reaches a critical value. The state of persistent activity below this transition consists of multiple stable periodic attractors, whose number increases at least as fast as the number of neurons in the network. At large shortcut density and for long enough delays the network dynamics exhibit exceedingly long chaotic transients, whose failure times follow a stretched exponential distribution. We show that this functional form arises for the ensemble-averaged activity if the failure time for each individual network realization is exponen- tially distributed

Reentrant and whirling hexagons in non-Boussinesq convection

We review recent computational results for hexagon patterns in non- Boussinesq convection. For sufficiently strong dependence of the fluid parameters on the temperature we find reentrance of steady hexagons, i.e. while near onset the hexagon patterns become unstable to rolls as usually, they become again stable in the strongly nonlinear regime. If the convection apparatus is rotated about a vertical axis the transition from hexagons to rolls is replaced by a Hopf bifurcation to whirling hexagons. For weak non-Boussinesq effects they display defect chaos of the type described by the two-dimensional (2D) complex Ginzburg-andau equation. For stronger non-Boussinesq effects the Hopf bifurcation becomes subcritical and localized bursting of the whirling amplitude is found. In this regime the cou- pling of the whirling amplitude to (small) deformations of the hexagon lattice becomes important. For yet stronger non-Boussinesq effects this coupling breaks up the hexagon lattice and strongly disordered states characterized by whirling and lattice defects are obtained.

Hexagonal patterns in a model for rotating convection

We study a model equation that mimics convection under rotation in a fluid with temperature- dependent properties (non-Boussinesq (NB)), high Prandtl number and idealized boundary conditions. It is based on a model equation proposed by Segel [1965] by adding rotation terms that lead to a Kuppers-Lortz instability [Kuppers & Lortz, 1969] and can develop into oscillating hexagons. We perform a weakly nonlinear analysis to find out explicitly the coefficients in the amplitude equation as functions of the rotation rate. These equations describe hexagons and os- cillating hexagons quite well, and include the Busse?Heikes (BH) model [Busse & Heikes, 1980] as a particular case. The sideband instabilities as well as short wavelength instabilities of such hexagonal patterns are discussed and the threshold for oscillating hexagons is determined.

Effect of a variable delay in delayed dynamical systems

We report numerical evidence of the effects of a periodic modulation in the delay time of a delayed dynamical system. By referring to a Mackey-Glass equation and by adding a modula- tion in the delay time, we describe how the solution of the system passes from being chaotic to shadow periodic states. We analyze this transition for both sinusoidal and sawtooth wave mod- ulations, and we give, in the latter case, the relationship between the period of the shadowed orbit and the amplitude of the modulation. Future goals and open questions are highlighted.

Coherent excitation of nonlinear oscillators

Coherently driven, dissipative nonlinear oscillators,(driving kept permanently in phase with the oscillators response) are proposed as systems with interesting dynamics. Results for simple, preliminary examples, which do not show chaotic behavior, are briefly discussed.

Use of Feynman diagrams in large-scale neural networks with nonlinear behaviour

A new proposal to the study of large-scale neural networks is reported. It is based on the use of similar graphs to the Feynman diagrams. A first general theory is presented and some interpretations are given. A propagator, based on the Green's function of the neuron, is the basis of the method. Application to a simple case is reported.

Quasi chaotic digital behaviour in an optically processing

Digital chaotic behavior in an optically processing element is reported. It is obtained as the result of processing two fixed train of bits. The process is performed with an Optically Programmable Logic Gate. Possible outputs for some specific conditions of the circuit are given. These outputs have some fractal characteristics, when input variations are considered. Digital chaotic behavior is obtained by using a feedback configuration. A random-like bit generator is presented.

Secure transmission of information by digital chaotic signals

Protecting signals is one of the main tasks in information transmission. A large number of different methods have been employed since many centuries ago. Most of them have been based on the use of certain signal added to the original one. When the composed signal is received, if the added signal is known, the initial information may be obtained. The main problem is the type of masking signal employed. One possibility is the use of chaotic signals, but they have a first strong limitation: the need to synchronize emitter and receiver. Optical communications systems, based on chaotic signals, have been proposed in a large number of papers. Moreover, because most of the communication systems are digital and conventional chaos generators are analogue, a conversion analogue-digital is needed. In this paper we will report a new system where the digital chaos is obtained from an optically programmable logic structure. This structure has been employed by the authors in optical computing and some previous results in chaotic signals have been reported. The main advantage of this new system is that an analogue-digital conversion is not needed. Previous works by the authors employed Self-Electrooptical Effect Devices but in this case more conventional structures, as semiconductor laser amplifiers, have been employed. The way to analyze the characteristics of digital chaotic signals will be reported as well as the method to synchronize the chaos generators located in the emitter and in the receiver.